Easy(?) field theory exercise: some silly mistake?

I've just been reading the chapter on roots of unity in
Joseph Rotman's book /Galois Theory/ (2nd ed. 1998).

I've been enjoying the book. There are a lot of errors,
but they are mostly typographical, and all minor, in that
they are easy to detect and correct, so they don't disturb
the reader's concentration. I'm not sure if what I've
just bumped into is another error, one which for the first
time has confused me; or if I'm just confused.

(My concentration has been poor for the last few days, so
I think I must be making some silly mistake - one which
I can't see, because my concentration is so poor!)

This chapter has only one exercise, which is to show that
the multiplicative group of an infinite field cannot be
cyclic.

But this seems weirdly simple to prove, given that the
statement of the exercise supplies two hints: (i) consider
separately the cases of characteristic 0 and characteristic
p > 0; (ii) in the latter case, consider separately the
cases of u transcendental and u algebraic over Z_p (where
u is the supposed generator of the group, i.e. F \ {0} =
<u>, where F is the given infinite field).

Anyway, here is the argument I have given; please shoot it
down, and make my day (not). :-)

Suppose F \ {0} = <u>. Consider the element -u. Because
F is infinite, -u is not equal to 1. Also u is not equal
to 0 (because 1 is not equal to 0), so unless F has char-
acteristic 2, we cannot have -u = u.

So there exists a positive integer n such that either -u
= u^{n+1} or -u = u^{-n}, i.e. u^n = -1 or u^{n+1} = -1.

But then u^m = 1, where m = 2n or m = 2n+2, and this would
make <u> finite, contrary to hypothesis.

By my reckoning, the only case that remains to be considered
is where F has characteristic 2.

Again by my (possibly confused) reckoning, the first case
in hint (ii) above simply doesn't arise, because we must
have (considering the element 1 + u, like we considered -u
before) for some positive integer n, either 1 + u = u^{n+1}
or 1 + u = u^{-n}, i.e. either u^{n+1} - u - 1 = 0 or
u^{n+1} + u^n - 1 = 0, and in either case u is algebraic of
degree at most n+1 over Z_2. And /that/ case can't arise
either, because it also makes F = Z_2(u) finite!

If it weren't for the hints, I would imagine that this is a
simple and valid solution of a simple exercise - but is it?

I'm convinced that as soon as I post this, I will see (or
worse, someone else will see) that I've made some complete
and utter howler; but I still can't see it, so here goes ...
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.

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